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Let L1 be the common tangent in the first quadrant to the circle x² + y² = 16 and the ellipse x²/25 + y²/4 = 1, and let d1 be the length of the segment of this tangent cut off between the coordinate axes. Which statement is correct?

1. d1 = 14/sqrt(3)
2. Equation of L1 is 2x + sqrt(3) y = 4*sqrt(7)
3. d1 = 4/sqrt(3)
4. Equation of L1 is x + sqrt(3) y = 4*sqrt(7)

Correct answer: d1 = 14/sqrt(3)

Solution

Tangency to the circle gives c² = 16(1+m²); tangency to the ellipse gives c² = 25 m² + 4. Equating: 25 m² + 4 = 16 + 16 m² -> 9 m² = 12 -> m² = 4/3. The intercept length between axes works out to 14/sqrt(3).

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